We show that similarly to the logarithmic mean-velocity profile in wall-bounded turbulence, the landscape topography presents an intermediate region with a logarithmic mean-elevation profile. Such profiles are present in complex topographies with channel branching and fractal river networks resulting from model simulation, controlled laboratory experiments, and natural landscapes. Dimensional and self-similarity arguments are used to corroborate this finding. We also tested the presence of logarithmic profiles in discrete, minimalist models of networks obtained from optimality principles (optimal channel networks) and directed percolation. The emergence of self-similar scaling appears as a robust outcome in dynamically different, but spatially bounded, complex systems, as a dimensional consequence of length-scale independence.

中文翻译：

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从湍流到景观：有限复杂系统中的对数均值轮廓

我们表明，与壁面湍流中的对数平均速度剖面相似，景观地形呈现了一个具有对数平均高度剖面的中间区域。这样的剖面存在于复杂的地形中，具有通道分支和分形河网，这些分流是由模型模拟，受控实验室实验和自然景观产生的。维度和自相似性参数用于证实这一发现。我们还测试了从最优性原理（最优通道网络）和定向渗滤获得的离散，极简网络模型中对数轮廓的存在。自相似缩放的出现似乎是在动态不同但受空间限制的复杂系统中的可靠结果，这是长度缩放独立性的维度结果。